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## Ideal Lenses and Mirrors

On this page we discuss our models of ideal lenses and mirrors, and say a little about how they differ from the real things.  (See the page on lens and mirror images for more information about the sorts of images each lens or mirror forms.)

### Properties of Any Ideal Lens or Mirror

An ideal lens or mirror has a couple of properties which are definitely not shared with real lenses and mirrors:
1. When a flat "scene" is projected through an ideal lens or reflected in an ideal mirror, a flat image is formed.  The flat image formed by a lens may appear on the opposite side of the lens from the original scene, as a "real" image, or it may be formed on the same side of the lens as the scene, in which case it's a "virtual" image.  The image formed by a mirror may appear in front of the mirror, in which case it's a "real" image, or may appear behind the mirror, as a "virtual" image.
In the case of a "real" image from either a lens or mirror, if you place a piece of paper in the plane of the image, it will be projected on the paper (i.e., it really is "real").  A "virtual" image, on the other hand, alway appears on the "other side" of the lens or mirror from the observer, and is viewed by "looking through" the lens or mirror.  You can see it, you can focus it and project it with another lens, but you can't just put a piece of paper in the plane of the image in order to project it -- it's not "real".

2. An ideal lens or mirror is itself flat and can be modeled as a uniform thin disk.
Note that we are assuming that simple lenses and mirrors form images, and we'll use that assumption later when we try to determine the details of their behavior.

### An Ideal Positive Lens

See figure 1.   In addition to properties (1) and (2) which are shared by any ideal lens or mirror, a positive (convex) ideal lens has the following properties:
• A ray entering parallel to the axis bends to pass through the focus of the lens (red rays in figure 1).
• A ray passing through the focus as it comes in bends so that it goes out parallel to the axis (light blue ray in figure 1).
• A ray passing through the center of the lens does not bend (green ray in figure 1).

 Figure 1: Ideal positive lens

### An Ideal Negative Lens

See figure 2.  In addition to properties (1) and (2) shared by any ideal lens or mirror, a negative (concave) ideal lens has the following properties:
• A ray entering along a line to the back focus bends to exit parallel to the lens axis (red rays in figure 1).
• A ray entering parallel to the axis bends so that it exits along a line through the front focus (pale blue ray in figure 1).
• A ray passing through the center of the lens does not bend (green ray in figure 1).
 Figure 2:  Ideal negative lens

### Ideal Mirrors

An ideal mirror is simply an ideal lens with a flat mirror embedded in it.

An ideal positive ("concave") mirror behaves like an ideal positive lens, except all paths are reflected about the Y axis on the line of the mirror.  In particular, figure 1 above describes the behavior of a concave mirror, if we "fold" the diagram in the middle, so the left and right foci are superimposed and all rays entering from the right leave to the right instead of the left.  In addition to properties (1) and (2), an ideal positive mirrors satisfies:
• A positive mirror has one focus, which lies on the axis of the mirror.
• Rays entering parallel to the axis are reflected on a line leading to the focus.  (Thus, rays coming in parallel to the axis are focused to a point, which is the "focus" of the mirror.)
• Rays which pass through the focus and strike the mirror are reflected parallel to the axis.
• Rays striking the center of the mirror are reflected just as they would be from a flat mirror, with angle of incidence equaling the angle of reflection.
Similarly, an ideal negative ("convex") mirror behaves like an ideal negative lens with rays being reflected upon entry, and we can use figure 2 above to describe it if we just "fold" it in the middle.  In addition to general properties (1) and (2) listed above, it has the following properties:
• A negative mirror has one focus, which lies on the axis, but behind the mirror.
• Rays entering parallel to the axis are reflected along a line leading from the focus.
• Rays entering on a line leading to the focus are reflected on a line parallel to the axis.
• Rays striking the center of the mirror are reflected just as they would be from a flat mirror, with angle of incidence equaling the angle of reflection.

### Paraboloidal Lenses and Mirrors

A physical lens or mirror which is ground to have a parabolic surface has neither of the first two "ideal" properties we listed.

Obviously, a real lens or curved mirror isn't flat (which was property 2).  (Fresnel lenses and mirrors come close but they have other issues which keep them from being "ideal".)

Less obvious is that a parabolic lens or mirror does not produce a flat image from a flat object (property 1).  It focuses a point source lying on its axis into a single point, as we have shown graphically, but extended objects are not focused into perfect images lying in plane; the farther from the axis the object extends, the less perfect is the image.

For a telescope mirror, this is usually irrelevant, as the angle of acceptance is typically tiny; over a sufficiently small angle, a parabolic mirror is almost ideal.

Camera lenses, on the other hand, have problems which push them rather far afield from our "ideal lens".  First and foremost, it's difficult to mass produce parabolic lenses, so camera lenses (the glass ones, at least) are typically constructed from spherical surfaces rather than paraboloids.  Spherical lenses are substantially worse at producing images than parabolic ones, and in fact don't even focus point sources lying on the axis to points.  If we view them as "bad paraboloids", a spheroidal lens may be described as having a varying focal length depending on how far from the axis you look -- the middle of the lens brings light to a focus at a different location from the edges of the lens.  This is termed "spherical aberration".  To compensate for that, lens designers use cascades of positive and negative lenses cleverly placed to clean up the focus.

Camera lenses have a more severe problem, though, which is that they are required to project an image onto a very wide field -- they need to bring things which are far off the axis to a sharp focus.  Furthermore, they need the image brightness to be the same across the whole field; even an "ideal" lens produces an image which is brighter in the middle and which falls off in brightness rather badly far from the axis.  Consequently, a perfectly ground simple parabolic lens would make a rather bad camera lens.

Camera lens design before the advent of computer modeling was extremely difficult, and was in fact as much art as science.

### Real Glass and Real Light

The index of refraction of real glass depends on the wavelength of the light, and real light consists of a spread of wavelengths.  Consequently a real lens made of real glass does not have a single focal length; instead, its focal length depends on the wavelength of the light being focused.  This is termed "chromatic aberration", and in a camera lens, the consequences may show up in a variety of symptoms.  In eyeglasses made of high-index high-dispersion plastic, one common consequence is that red objects appear either behind or in front of blue objects when they actually lie in the same plane (an effect with which I'm extremely familiar).

Camera lens designers use glass made of a variety of materials with varying "dispersion", and each camera lens is actually built from a cascade of simple lenses in combination which (are supposed to) cancel out chromatic aberration; that is not a simple subject and is beyond the scope of what we're covering in this section.

### An Aside:  Inexpensive Plastic Magnifying Glasses

Since we may be talking about experiments one can do with lenses (some time in the future!), it's worth saying a few words about plastic lenses and cheap magnifying glasses.

In mass production, glass is difficult to grind into anything other than spherical surfaces, and in fact is rather expensive even to grind at all.  Plastic, on the other hand, is inexpensive to mold, and can be molded into any (possibly aspheric) surface desired, no grinding needed.  This makes it possible to produce plastic lenses far more cheaply than glass lenses.

Unfortunately, while molded plastic lenses are cheap, they also can be extremely poor.  The problem is the plastic doesn't always hold its shape as it cools after it's removed from the mold (typically it shrinks unevenly).  Consequently, cheap plastic magnifiers are often so poor quality that they can't be used for any kind of optical experiments -- they fail to form coherent images when scenes are projected through them.  For fiddling with lenses, it's worth either buying some for the purpose, or at least finding some high-quality magnifying glasses.

Page created on 09/16/2007